STOCHASTIC SPACE TRANSFORMS IN SUBSURFACE HYDROLOGY .2. GENERALIZED SPECTRAL DECOMPOSITIONS AND PLANCHEREL REPRESENTATIONS

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators red uce the original multi-dimensional problem to the one-dimensional spac e, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the pres ent work we discuss developments in the theoretical formulation of flo w models with space-dependent coefficients in terms of space transform ations. The formulation is based on stochastic Radon operator represen tations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tra ctable expressions of the space transformed flow equation. A Planchere l representation of the space transformation product of the head poten tial and the log-conductivity is also obtained. A test problem is firs t considered in detail and the solutions obtained by means of the prop osed approach are compared with the exact solutions obtained by standa rd partial differential equation methods. Then, solutions of three-dim ensional groundwater flow are derived starting from solutions of a one -dimensional model along various directions in space. A step-by-step n umerical formulation of the approach to the flow problem is also discu ssed, which is useful for practical applications. Finally, the space t ransformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.